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Properties

Label 2-6120-17.16-c1-0-71
Degree 22
Conductor 61206120
Sign 0.312+0.949i0.312 + 0.949i
Analytic cond. 48.868448.8684
Root an. cond. 6.990596.99059
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·5-s − 1.71i·7-s − 1.71i·11-s + 4.20·13-s + (3.91 − 1.28i)17-s + 3.33·19-s − 6.20i·23-s − 25-s − 3.71i·29-s − 3.62i·31-s + 1.71·35-s − 3.71i·37-s + 10.9i·41-s + 1.15·43-s − 9.17·47-s + ⋯
L(s)  = 1  + 0.447i·5-s − 0.646i·7-s − 0.515i·11-s + 1.16·13-s + (0.949 − 0.312i)17-s + 0.765·19-s − 1.29i·23-s − 0.200·25-s − 0.689i·29-s − 0.651i·31-s + 0.289·35-s − 0.610i·37-s + 1.71i·41-s + 0.176·43-s − 1.33·47-s + ⋯

Functional equation

Λ(s)=(6120s/2 ΓC(s) L(s)=((0.312+0.949i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(6120s/2 ΓC(s+1/2) L(s)=((0.312+0.949i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 61206120    =    23⋅32⋅5⋅172^{3} \cdot 3^{2} \cdot 5 \cdot 17
Sign: 0.312+0.949i0.312 + 0.949i
Analytic conductor: 48.868448.8684
Root analytic conductor: 6.990596.99059
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ6120(1801,⋅)\chi_{6120} (1801, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 6120, ( :1/2), 0.312+0.949i)(2,\ 6120,\ (\ :1/2),\ 0.312 + 0.949i)

Particular Values

L(1)L(1) ≈\approx 1.9988616061.998861606
L(12)L(\frac12) ≈\approx 1.9988616061.998861606
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
5 1−iT 1 - iT
17 1+(−3.91+1.28i)T 1 + (-3.91 + 1.28i)T
good7 1+1.71iT−7T2 1 + 1.71iT - 7T^{2}
11 1+1.71iT−11T2 1 + 1.71iT - 11T^{2}
13 1−4.20T+13T2 1 - 4.20T + 13T^{2}
19 1−3.33T+19T2 1 - 3.33T + 19T^{2}
23 1+6.20iT−23T2 1 + 6.20iT - 23T^{2}
29 1+3.71iT−29T2 1 + 3.71iT - 29T^{2}
31 1+3.62iT−31T2 1 + 3.62iT - 31T^{2}
37 1+3.71iT−37T2 1 + 3.71iT - 37T^{2}
41 1−10.9iT−41T2 1 - 10.9iT - 41T^{2}
43 1−1.15T+43T2 1 - 1.15T + 43T^{2}
47 1+9.17T+47T2 1 + 9.17T + 47T^{2}
53 1+7.91T+53T2 1 + 7.91T + 53T^{2}
59 1+13.4T+59T2 1 + 13.4T + 59T^{2}
61 1−1.79iT−61T2 1 - 1.79iT - 61T^{2}
67 1−6.78T+67T2 1 - 6.78T + 67T^{2}
71 1+8.41iT−71T2 1 + 8.41iT - 71T^{2}
73 1+6.38iT−73T2 1 + 6.38iT - 73T^{2}
79 1−7.15iT−79T2 1 - 7.15iT - 79T^{2}
83 1+1.83T+83T2 1 + 1.83T + 83T^{2}
89 1+3.62T+89T2 1 + 3.62T + 89T^{2}
97 1−4.84iT−97T2 1 - 4.84iT - 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.925554241376838730206169988472, −7.29921194363174030617371002043, −6.31862491568440972829634867358, −6.06697849281670078738177485035, −5.01794581996673426890631286445, −4.19013168786127387472203383633, −3.40633269478720327528702930771, −2.81111864604084126220039669047, −1.50070477321755276050712810507, −0.56176115867467754822769772809, 1.17293833605922887236333773399, 1.82153566648206638954955062258, 3.15234148749533175173720939666, 3.63204252411202958112971648551, 4.70799489130361158426982311686, 5.45264880992298007939741475216, 5.87605750443466502152355512686, 6.82163405740894771858389844600, 7.57251856737053927021568291911, 8.239898402909554763568250450405

Graph of the ZZ-function along the critical line